I believe I would have read this in 1988, as a stand-alone paperback novel. (I was avoiding all, or at least most, series that summer, having gotten fed up with David Eddings.)
I don't really recall a lot of the story, except specifically that The Hero (male) Didn't Get the Girl in the end. I recall that the protagonist was a musician-type guy with some kind of pointy weapon that he used when his magic music wasn't enough. (Which was pretty often, especially at the start, IIRC.) I don't think he was actually a "bard;" he simply played an instrument and had the ability to make magic with music. I'm not sure what instrument he played, but it was something portable (like a flute or a lyre) and pre-Renaissance (so not a violin). He came from some town and hadn't really traveled much before the story began.
He mostly traveled alone, met a few people along the way but never really formed a party.
At one point he rescued a (female) forest spirit of some fashion. (This was one occasion when he had to use the sharp pointy thing not his music.) She wasn't human, but she wasn't an anthropomorphic animal either; she belonged to some kind of nature-connected magic race like a dryad (but not an elf). I think he kind of started to fall in love with her, but her boyfriend showed up and she was like "Cya!"
My memory is really vague about the ending; I don't remember how he meets up with the Damsel in Distress or how the final confrontation with the Big Bad went down, but I think she beat up the Big Bad and then basically told the hero "KKThxBai!"
It's definitely not Spellsinger; Jon-Tom arrived from our world (he wasn't a native); he met anthropomorphic animals (Mudge, Clothahump), formed a party... and he does get the girl, just not in the first book of the series.